We give the first algorithmic study of a class of “covering tour”
problems related to the geometric Traveling Salesman Problem:
Find a polygonal tour for a cutter so that it sweeps
out a specified region (“pocket”), in order to minimize a cost that
depends not only on the length of the tour but also on the number of
turns. These problems arise naturally in manufacturing
applications of computational geometry to automatic tool path
generation and automatic inspection systems, as well as arc routing
(“postman”) problems with turn penalties. We prove lower bounds
(NP-completeness of minimum-turn milling) and give efficient
approximation algorithms for several natural versions of the problem,
including a polynomial-time approximation scheme based on a novel
adaptation of the m-guillotine method.